750 10. THE CASE LE .Cj(G,T) NOT NORMAL IN M.By B.4.8.2, A= R 1 and rz Q, A= 1, so by B.4.8.4, Zq = V1CzQ(L). This shows
ZQ = VCzQ (L 0 ), R = J(T)Q, and Cr(v 2 ) = (T n L)R2Q, establishing (4) except
for its final assertion. Notice it also shows Zn V :::; Zr 1 n V :::; Cv(Lo). But
T1 = ToQ, so CzQ (Lo) :::; Zr 1 • Conversely, Zr 1 :::; Zq and we saw VnZr 1 :::; Cv(Lo),
so Zr 1 :::; CzQ (L 0 ), and hence (2) holds. Further Z :::; Zr 1 , so (2) implies (1).
Finally Q:::; Cr(v 2 ) and Q = F*(L 0 T), so D1(Z(Cr(v2))):::; ZQ = VZr 1 ; therefore
D1(Z(Cr(v2))) = Zr 1 C~(Cr(v2)) = Zr 1 (v2), completing the proof of (4), and
hence of the lemma. D
We are now in a position to produce a crucial bound on the weak closure
parameter r of Definition E.3.3:
PROPOSITION 10.2.9. (1) Ca(v) _:::; M for each v E Vt.
(2) r(G, V) ~ m(v;).
(3) If v E Vi -Cv,(Lo), then Ca(v):::; NM(Vi).
PROOF. Part (3) follows from (1) and the fact that M permutes {Vi, Vz} and
Vin Vz = Cv(Lo). Also (1) implies (2), so it remains to prove (1).
Let v E vt, and suppose by way of contradiction that H := Ca(v) i M.
Without loss Tv := Cr(v) E Sylz(CM(v)). By 10.2.6.1, v t/:. Cv 2 (LoT1).
We claim first that Na(Tv) :::; M. If J(T) :::; Cr(V), this follows from 3.2.10.8;
so by 10.1.2.1 we may assume that one of the first three cases of 10.1.1 holds.
Suppose first that case (3) of 10.1.1 holds, and also Cvi(L) =/= 1. Then by 10.2.8.2,
Zr 1 := D1(Z(T 1 )) ~ Cv(L 0 ), so v tj:. Cv(Lo) using our observation in the previous
paragraph. Therefore as L 2 is transitive on flt, we may assume ( v) = Cv- 2 (T1).
Hence by 10.2.8.4, Tv = (T n L)R2Q, and Zv := D1(Z(Tv)) = Zr 1 (v). By 10.2.8.1,
Lo centralizes Z, so Cc(Zv) :::; Ca(Z) :::; M = !M(L 0 T), and hence by 10.1.3, L
is the unique member of C(Ca(Zv)) of order divisible by 3. Therefore Na(Tv) :::;
Na(Zv) :::; Na(L) :::; M using 10.2.6.2. We now turn to the remaining subcase
of case (3) of 10.1.1, where CVi (L) = 1. Then Tv = Ti, so Nc(Tv) :::; M by
10.2.5. Finally in cases (1) and (2) of 10.1.1, S :::; T 1 by 10.1.2.6; and in case (2),
S centralizes both singular and nonsingular vectors. So in either case, S :::; Tv.
Therefore S = Baum(Tv) and Na(Tv) :::; Na(S)) :::; M by 10.2.1. This completes
the proof of the claim.
As Na(Tv) :::; M by the claim, while we chose Tv E Sylz(CM(v)), Tv E Syl2(H).
Also L :::; H, so by 1.2.4, L :::; I E C(H), with I ::;! H by ( +) in 1.2.4. By 10.2.6,
Na(L):::; M, so L <I and hence Ii M. Thus I is described in A.3.12.
Suppose first that I is quasisimple. Then V 1 n Z(I) :::; Cv 1 (L), so V 1 ~
Vi/Cv 1 (L) is a subquotient of R 2 (LZ(I)/Z(I)). Inspecting the list in A.3.12 for
embeddings with such a subquotient appearing in 10.1.1, we conclude that case (1)
or (3) of 10.1.1 holds; and keeping in mind that Na(V 1 ) :::; M so that L ::::) N 1 (V 1 ),
we conclude that either:
(i) L ~ L 2 (2n), and either I/Z(I) is of Lie type and Lie rank 2 over F 2 n, or
n = 2 and I/Z(I) is M22, M22, or M23; or
(ii) case (3) of 10.1.1 holds with CVi (L) :::; Z(I), and I/Z(I) is L 4 (2), L 5 (2),
M24, J4, HS, or Ru.
In particular either Cr(L) = Cr(I), or I.~ Sp 4 (2n) in (i), using I.1.3 to conclude
the Schur multiplier of Sp 4 (2n) is trivial when n > 1. When Cr(L) = Cr(I),